Statement -1:If
`e^{("sin"^(2)x + "sin"^(4)x + "sin"^(6)x +…)"log"_(e)2] ` satisfie the equation `x^(2)-9x +8=0`, "then " `(cosx)/(cosx + sinx) = (sqrt(3)-1)/(2), 0 lt x lt (pi)/(2)`.
Statement-2: The sum `sin^(2) x + sin^(4)x + sin^(6) x + .... oo` is equal to `tan^(2)x`

Options

A. Statement -1 is true, Statement-2 is true, Statement -2 is a correct explanation for Statement-1.
B. Statement -1 is True, Statement-2 is True, Statement -2 is not a correct explanation for Statement -1.
C. Statement-1 is True, Statement-2 is False.
D. Statement -1 is False, Statement-2 is True.

Answer

A

Explanation

We have,
`"sin"^(2)x + "sin"^(4)x + "sin"^(6)x + … "in"f= ("sin"^(2)x)/(1-"sin"^(2)x) = "tan"^(2) x`
So, statement -2 is true.
Using statement -2, we have
`"exp"{"(sin"^(2)x + "sin"^(4)x + "sin"^(6)x +…)"log"_(e)2]`
` =e^(("tan"^(2)x) "log"_(e)2) = 2^("tan"^(2)x)`
It is given that `2^("tan"^(2)x)` satisfies the equation `x^(2) -9x + 8=0`
`therefore 2^("tan"^(2)x) = 2^(3) "or" 2^("tan"^(2)x) = 1`
`rArr "tan"^(2) x = 3 "or tan"^(2)x =0`
`rArr "tan" x = sqrt(3) rArr x = (pi)/(3) " " [because 0 le x le (pi)/(2)]`
`therefore ("cos"x)/("cos" x + "sin"x) = (1)/( 1 +"tan"x) = (1)/(1+sqrt(3)) = (sqrt(3)-1)/(2)`
So, statement-1 is true. Also, statement -2 is a correct explanation for statement-1.